Evaluate 18/(4+ square root of 7)-5 square root of 7

$\frac{18}{4 + \sqrt{7}} - 5 \sqrt{7}$

Step 1

Simplify each term.

Multiply $\frac{18}{4 + \sqrt{7}}$ by $\frac{4 - \sqrt{7}}{4 - \sqrt{7}}$ .

$\frac{18}{4 + \sqrt{7}} \cdot \frac{4 - \sqrt{7}}{4 - \sqrt{7}} - 5 \sqrt{7}$

Multiply $\frac{18}{4 + \sqrt{7}}$ by $\frac{4 - \sqrt{7}}{4 - \sqrt{7}}$ .

$\frac{18 (4 - \sqrt{7})}{(4 + \sqrt{7}) (4 - \sqrt{7})} - 5 \sqrt{7}$

Expand the denominator using the FOIL method.

$\frac{18 (4 - \sqrt{7})}{16 - 4 \sqrt{7} + \sqrt{7} \cdot 4 - {\sqrt{7}}^{2}} - 5 \sqrt{7}$

Simplify.

$\frac{18 (4 - \sqrt{7})}{9} - 5 \sqrt{7}$

Cancel the common factor of $18$ and $9$ .

Factor $9$ out of $18 (4 - \sqrt{7})$ .

$\frac{9 (2 (4 - \sqrt{7}))}{9} - 5 \sqrt{7}$

Cancel the common factors.

Factor $9$ out of $9$ .

$\frac{9 (2 (4 - \sqrt{7}))}{9 (1)} - 5 \sqrt{7}$

Cancel the common factor.

$\frac{9 (2 (4 - \sqrt{7}))}{9 \cdot 1} - 5 \sqrt{7}$

Rewrite the expression.

$\frac{2 (4 - \sqrt{7})}{1} - 5 \sqrt{7}$

Divide $2 (4 - \sqrt{7})$ by $1$ .

$2 (4 - \sqrt{7}) - 5 \sqrt{7}$

Apply the distributive property.

$2 \cdot 4 + 2 (- \sqrt{7}) - 5 \sqrt{7}$

Multiply $2$ by $4$ .

$8 + 2 (- \sqrt{7}) - 5 \sqrt{7}$

Multiply $- 1$ by $2$ .

$8 - 2 \sqrt{7} - 5 \sqrt{7}$

Step 2

Subtract $5 \sqrt{7}$ from $- 2 \sqrt{7}$ .

$8 - 7 \sqrt{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$8 - 7 \sqrt{7}$

Decimal Form:

$- 10.52025917 \dots$